On a lightly floured surface roll dough to a 14-inch circle. Sprinkle an extra-large baking sheet with cornmeal. Transfer dough the the baking sheet. Prick all over with a fork. Use the baking sheet to transfer dough to the greased rack of a grill directly over medium-high heat. Cover and grill 3 to 4 minutes or until bottom is brown, rotating halfway though. Turn crust and transfer to baking sheet. Top with 1/2 cup dressing, mozzarella cheese, Parmesan cheese, and black pepper. Return to grill. Cover and grill 3 to 4 minutes more or until bottom is brown and cheese is melted, rotating once halfway through. Remove from grill.
Immediately top with romaine, chicken, tomatoes, and onion. Drizzle with remaining 1/2 cup dressing. Garnish with additional shredded mozzarella and/or shaved Parmesan, if desired.